The univalent Bloch-Landau constant, harmonic symmetry and ...
The univalent Bloch-Landau constant, harmonic symmetry and ...
The univalent Bloch-Landau constant, harmonic symmetry and ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
the extremal configuration will be <strong>harmonic</strong>ally symmetric at infinity.<br />
As noted in the introduction, this latter observation was first made by<br />
Lavrentiev [15].<br />
Wework withdomains Ω = Ω z0 ,R asshown inFigure4, where R > 3,<br />
|z 0 | < R 3 , <strong>and</strong> the arc γ z0 is chosen so that Ω z0 ,R is <strong>harmonic</strong>ally<br />
symmetric in γ z0 with respect to 0. If g is a conformal map of the unit<br />
7<br />
γ z0<br />
Ω z0 ,R z 0<br />
−8 1<br />
z 0<br />
R 3<br />
γ z0<br />
Figure 4. A domain Ω z0 ,R<br />
disk D onto such a domain Ω z0 ,R, with g(0) = 0, then f(z) = z 3 √<br />
g(z 3 )<br />
z 3<br />
is a conformal map of D onto a domain U w,R as shown in Figure 5.<br />
<strong>The</strong> arcs that appear are all <strong>harmonic</strong>ally symmetric, <strong>and</strong> thus the<br />
w<br />
U w,R<br />
−2 1<br />
R<br />
w<br />
Figure 5. A domain U w,R<br />
conformal mapping of the unit disk onto U w,R is a good c<strong>and</strong>idate for<br />
having a relatively large derivative at the origin. This derivative is<br />
|f ′ (0)| = 3√ |g ′ (0)|. In order that this provide a useful estimate of the<br />
<strong>Bloch</strong>-<strong>L<strong>and</strong>au</strong> <strong>constant</strong>, we need to arrange for U w,R to have inradius<br />
1. We leave this aside for the moment <strong>and</strong> show how to use Fedorov’s<br />
results on capacity to compute |f ′ (0)| for given z 0 <strong>and</strong> R. We write<br />
k for the Koebe mapping k(z) = z/(1 −z) 2 of the unit disk onto the<br />
plane slit along the negative real axis from minus infinity to −1/4.